Technical Field
The present disclosure relates to methods of reducing the swell potential of an expansive clayey soil. More specifically, the present disclosure relates to methods of reducing the swell potential of an expansive clayey soil comprising at least one expansive clay mineral with the aid of a nano-level constitutive modeling and preferably with a swelling reduction agent selected from calcite, gypsum, potassium chloride, a composition comprising exchangeable K+, a composition comprising exchangeable Ca2+, a composition comprising exchangeable Mg2+, and a combination thereof.
Description of the Related Art
The “background” description provided herein is for the purpose of generally presenting the context of the disclosure. Work of the presently named inventors, to the extent it is described in this background section, as well as aspects of the description which may not otherwise qualify as prior art at the time of filing, is neither expressly nor impliedly admitted as prior art against the present invention.
Expansive clays are widely prevalent all over the world as one of the most problematic and challenging soils. These soils undergo significant volume change with the change in the moisture regime, thereby posing problems to the stability of the structures founded on such strata. The expansive clays become highly erratic in behavior especially when present in unsaturated/partially saturated state having fluctuations of the saturation levels. More challenging is the fact that foundations of most civil engineering structures are generally placed in the partially saturated soil zones with a continuously varying degree of saturation with the environmental and weather conditions. The American Society of Civil Engineers (ASCE, 2013) estimates that 25% of all the homes in the United States suffer some extent of damage by expansive soils and an estimate shows that in a typical year in the United States these soils cause a financial loss to property owners greater than other natural disasters such as earthquakes, floods, hurricanes and tornadoes combined. Expansive soils are also commonly present in the Kingdom of Saudi Arabia and concentrate mostly in the populated cities. These expansive soil deposits present in the Kingdom of Saudi Arabia contain high percentages of expansive clay minerals (Table 1). High percentages of the expansive clay minerals result in high to very high swell potential of these soil deposits (Table 2). Consequently, structural and functional damage to the structures of the entire housing complexes by expansive clays is quite common in several areas of KSA. Empirical and experimental based solutions and formulae to predict the expansive potential of these soils have not been able to provide a comprehensive understanding for the various possible variations in the fabric and structure of the natural and compacted expansive clay soils.
TABLE 1Mineralogical analysis of expansive clay deposits in the Kingdom of Saudi Arabia(Hameed, 1991).Sample No.LocationBH/TP No.Depth (m)Mineral type (% composition)1Al-Khars, Al-HasaBH-92.5-2.7C(50), Q(10), P(8), K(5), I(9), S(7)2Mahasen-Aramco, Al-HasaBH-13  20-2.25C(34), Q(16), P(8), K(3), I(31), S(4), St(4)3Al-HamadiyaBH-111.0-1.3C(35), Q(10), P(8), K(3), I(30), S(8), St(5)4Al-SalehiyaBH-122.4-2.7C(19), Q(19), P(4), K(4), I(47), D(1), S(8)5Al-Khars, Al-HasaTP-71.1C(39), Q(11), P(10), K(11), I(19), S(6), St(4)6Al-Naathel, Al-HasaTP-112.0-2.2C(61), Q(24), P(6), K(3), S(6)7Mahasen-Aramco, Al-HasaTP-112.0-2.2C(27), Q(19), P(10), K(5), I(32), S(4), St(3)8Housing AreaBH-13.8Q(9), P(9), S(30), St(12), T, I(40)9Housing AreaBH-31.8-2.1Q(7), K(<1), S(25), St(10), D(32), I10Umm Al-SabekBH-60.45-0.6 Q(4), K(<1), P(2), S(13), G(11), D(48), I(22)11Umm Al-HammamBH-8 5.6-5.75C(<1), Q(10), P(11), D(<1), S(39), I(39)C = CalciteQ = QuartzP = PalygorsliteSt = SepioliteS = SmectiteK = KaolimiteI = IlliteD = DolomiteG = GypsumT = Talc
TABLE 2Geotechnical properties of the expansive clay deposits in Qatif area,Kingdom of Saudi Arabia (Dafalla and Shamrani, 2012).Brown calcareous green andProperty range forbrown clay, symbol (USCS):Al Qatif soilsCH and MHAvg. min.Avg. max.Dry unit weight, gd, kN/m31014Water contet, wn, %1540Liquid limit, LL, %120160Plastic limit, PL, %3060Plasticity index, PI, %90100Shrinkage limit, SL, %915Percent sand, %05Percent silt, %1550Percent clay, %5090Specific gravity, Gs2.52.6Selling pressure, kN/m32001000Swell percent220
Studies of the interaction of clay minerals with pore fluids and their contributions to the fabric, structure, and macroscale behavior and properties of clay minerals are critical not only in the fields of geotechnical engineering but also in geoenvironmental engineering, material sciences, pharmaceutical sciences etc. See Katti, D. R., Katti, K. S., Amaasinghe, P. M. and Pradhan, S. M. (2011), “An insight into role of clay-fluid molecular interactions on the microstructure and Macroscale properties of swelling clay”, Alonso and Gens (eds), Unsaturated Soils, 2011 Taylor and Francis Group, London, incorporated herein by reference in its entirety. Since the emergence of the unsaturated geotechnical engineering, performance of numerical modeling of the realistic volume change behavior of the expansive clays is a challenge for the geotechnical engineers. Consequently, several efforts have been made to develop constitutive models for the behavior of the expansive soils by performing parametric studies mostly at the macro behavior level and to a lesser extent at a molecular level. All the developed constitutive models do not comprehensibly incorporate the coupling of the behavior at the macro, micro, and nano/molecular levels. Moreover, most of the developed constitutive models pertain to the standard expansive clay minerals compacted under controlled conditions; models covering the natural and real soil fabric do not exist.
Lack of proper understanding and knowledge of the nano/molecular level interactions of the clay minerals with pore fluids and the other non-swelling constituents have limited the development of specific constitutive models encompassing the accurate behavior under several possible combinations of clay, fluid and other non-swelling particles. This behavior becomes further complex for the swelling clays when the interaction between clay, fluid, and the non-swelling clay particles become predominant.
The swelling behavior of expansive soils is intrinsically controlled by their natural fabric and structure. Although the fabric of expansive soils is quite complex, Mitchell attempted to discretize soil fabric to be consisting of three general regimes of elementary particle arrangements as single form of particle interaction at the level of individual clay, silt, or sand particles, particle assemblage as units of particle organization having definable physical boundaries and pore spaces as fluid and/or gas filled voids within the soil fabric. See Mitchell, J. K. (2005), “Fundamentals of soil behaviour”, 3rd Edition, John Wiley and Sons, Inc., New York, incorporated herein by reference in its entirety. Mitchell divided the fabric of a soil into three levels of scale as microfabric, minifabric, and macrofabric. Microfabric is defined as regular aggregations of particles and the very small pores between them; typical fabric units are up to a few tens of micrometers across. The minifabric consists of the aggregations of the microfabric and the interassemblage pores between them; minifabric being a few hundred micrometers in size. Finally, macrofabric may contain cracks, root holes, laminations, and the like that correspond to the transassemblage pores. Abduljauwad and Al-Sulaimani carried out detailed research on the swelling potential of the clay soils in Qatif area of Saudi Arabia. See Abduljauwad, S. N. and Al-Sulaimani, G. J. (1993), “Determination of Swell potential of Al-Qatif Clay”, Geotechnical Testing Journal, ASCE, December 1993, pp. 469-484, incorporated herein by reference in its entirety. As a result of these studies, they found substantial differences in the swelling potential assessed from the laboratory conventional Oedometer tests, laboratory tests on large scale block samples, and field tests on the subsurface strata (Table 3). They attributed these differences to the contribution of several macro to nano level structural features that might have been masked in the small scale laboratory tests. Moreover, El Sohby and Rabba also showed that swell percentage and pressure does not have a linear relationship with the various percentages of sand and silt content (FIGS. 1A and 1B). See El Sohby, M. A. and Rabba, E. A. (1981), “Some Factors affecting the swelling of clayey soils”, Geotechnical Engineering, Vol 12, page 19-39, incorporated herein by reference in its entirety.
TABLE 3Comparison of swelling potential based on the results from thelaboratory and field tests (Abduljauwad and Al-Sulaimani, 1993).Percentage ofSwellingHeave,MethodSwell, %Pressure kPammOedometerImproved simple oedometer36310063.9Constant volume—80047.5Reverse curve8.8200058.6Suction——36.8Triaxial14.342039.6Simulation swelling test15.0—37.4Field15.418038.4
Based on the premise by Mitchell and the conclusions of Abduljauwad et al. and El Sohby and Rabba, it seems that in addition to macro to nano level behavior, macro to nano level features should also be considered in the constitutive modeling of expansive clays. See Abduljauwad, S. N., Al-Sulaimani, G. J., Basunbul, I. A., and Al-Buraim, I. (1998), “Laboratory and field studies of response of structures to heave of expansive clay”, Geotechnique, 48(1): 103-121, incorporated herein by reference in its entirety. Gens and Alonso presented a mathematical model for the expansive clays. See Gens, A. and Alonso, E. E. (1992), “A framework for the behavior of unsaturated expansive clays”, Canadian Geotechnical Journal 29, 1013-1032 (1992), incorporated herein by reference in its entirety. Lumped fabric consisting of micro and nano level pores and idealization of a single mineral fabric attained under controlled compaction conditions by Gens and Alonso and their followers might not have led to the formulation of a complete representative behavior model.
Nano or molecular level processes may play a role in understanding the volume change behavior of expansive clays. Some studies have been conducted to simulate the swelling and/or water absorption behavior of the single or isolated expansive clay minerals, but modelling of the real/natural expansive soil fabric and its interaction with pore fluids at molecular level is still lacking. Moreover, no efforts have been directed to couple the macro and micro scale material behavior based on the findings of these molecular simulations. As molecular level modeling studies could lead to the real insights into soil behavior, it would result in the validation and/or modifications of several macroscopic (continuum) constitutive behaviors. Recent advances in numerical computational methods, high performance hardware, molecular modeling software, and experimental techniques could be used to provide the real insight into the real behavior at the molecular level.
Attempts to predict the expansive or swell potential of expansive clay minerals or expansive clayey soils comprising expansive clay minerals have not been able to provide a comprehensive understanding for the various possible variations in the fabric and structure of the natural and compacted expansive clayey soils. Since the emergence of the unsaturated geotechnical engineering, performance of numerical modeling of the realistic volume change behavior of the expansive clays is a challenge for the geotechnical engineers. Consequently, several efforts have been made to develop constitutive models for the behavior of the expansive soils by performing parametric study mostly at the macro behavior level and to quite lesser extent at molecular level. All the developed constitutive models do not comprehensibly incorporate the coupling of the behavior at the macro, micro, and nano levels. Moreover, most of the developed constitutive models pertain to the standard expansive clay minerals compacted under controlled conditions; models covering the natural and real soil fabric do not exist.
Lack of proper understanding and knowledge of the molecular and nano level interactions of the clay minerals with the pore fluids and the other non-swelling constituents have limited the development of specific constitutive models encompassing the accurate behavior under several possible combinations of clay, fluid and other non-swelling particles. This behavior becomes further complex for the swelling clays when the interaction between clay, fluid, and the non-swelling clay particles become predominant.
The present disclosure covers the general comprehension of the fabric and structure of the swelling clays, swelling mechanism, and the corresponding level of efforts in the constitutive and molecular level modeling of expansive clayey soils. All these pertinent issues related to expansive clays are discussed in detail herein.
The excessive volume change tendency of expansive clays is mainly attributed to the presence of expansive clay minerals in the soil fabric. These expansive clay minerals have got high affinity to the water and dissolved ions due to the net unbalanced electrical charges present on their surfaces. Volume change of the clay structure occurs once these expansive minerals absorb water and move from one partially saturated state to another. The volume change behavior is invariably controlled by many factors including type of clay minerals, current degree of saturation, past wetting-drying cycles, fabric and structure created during the compaction/natural deposition, presence of non-expansive minerals, their sizes, percentages and distribution in the matrix. A comprehensive constitutive model should encompass all these factors and their relative contribution to the physico-chemical-mechanical interactions at various scale levels. In order to integrate all these factors in a constitutive model, understanding the behavior of the fabric at micro and nano level and its association with the macro behavior is required.
Most of the expansive clay minerals belong to the Smectite group and their typical expandable structure consisting of alternate silicate and alumina sheets is shown in FIG. 2 (Mitchell). Each clay particle could be conceptualized as a flake/sheet like structure having dimensions of an order of nanometer with a length or width to thickness ratio of about 2000:1. See Sharma, R. S. (1998), “Mechanical Behavior of Unsaturated Highly Expansive Clays”, PhD Thesis, Oxford University, UK, incorporated herein by reference in its entirety. The clay particles are also referred to as particles, lamellae or micelles at these finest levels. See Quirk, J. P. and Murray, R. S. (1991), “Towards a model for soil structural behavior”, Australian Journal of Soil Research 29, 829-867; Oades, J. M. and Waters, A. G., (1991), “Aggregate hierarchy in soils”, Australian Journal of Soil Research, 29, 815-828, each incorporated herein by reference in their entirety. Isomorphous substitution, broken edges, and eccentric positive and negative charge centers result in net unbalanced charges on these particles. As a result of these unbalanced charges, these clay particles or sheets combine to form platelets of each about ten sheets (100:1) (Oades and Waters). These have also been called grains, crystals or quasi-crystals (Quirk and Murray). Various bonding forces ranging from hydrogen bonds in kaolinite to van der Waals and cation bonding in montmorillonite exist in the individual clay particles or sheets. The group of platelets are present as micro-aggregates (Oades and Waters) and clusters at the microscopic level and peds, macro-aggregates (Oades and Waters) or prisms at macro level. See Thomasson, A. J. (1978), “Towards an objective classification of soil structure”, Journal of Soil Science 29, 38-46; Cabidoche, Y. M. and Ruy, S. (2001), “Field shrinkage curves of a swelling clay soil: analysis of multiple structural swelling and shrinkage phases in the prisms of a vertisol”, Australian Journal of Soil Research 39, 143-160, each incorporated herein by reference in their entirety.
Lambe provided a conceptual picture of the clay fabric, although his work was mainly related to the compacted clays only. See Lambe, T. W. (1958), “The structure of compacted clay”, Journal of Soil Mechanics and Foundations Division, ASCE, Vol 84 (SM2), 1654, incorporated herein by reference in its entirety. He defined the bimodal fabric on dry side and massive and unimodal fabric on wet side of the optimum with the microvoids in between platelets and macrovoids in group of platelets. In his models, he identified three levels of fabric corresponding to three levels of void/fluid filled spaces, intra-platelet spaces between individual unit layers, small voids (microvoids) between individual clay platelets between larger flocs of packets of soils, and macrovoids between larger flocs and packets.
Gens and Alonso, considered as pioneers in formulating the first constitutive model framework for the expansive soils, envisaged an expansive clay fabric (FIGS. 3A, 3B and 3C). They conceptualized the structural arrangement consisting of three basic microfabric features: elementary particle arrangements or quasi-crystals, particle assemblages, and pore spaces. Gens and Alonso described the particle assemblages formed by arrays of elementary particle arrangements as matrices. In their model, pore spaces in the matrices are made up of intramatrix pores existing between elementary particle arrangements. Elementary particle arrangements join together to make aggregates resulting in a three-dimensional structure of a granular type. Both inter and intra-aggregate pore spaces exist in the aggregated structure. A further level of void space also exists in the intraelement pores separating the clay platelets in the elementary particle arrangements. They related both the expansive and collapse type of phenomena to these forms of fabric. Clay structure conceptualized by Gens and Alonso was further supported by SEM micrographs of clay samples at optimum and dry and wet sides of optimum by Delage and Graham. See Delage, P. and Graham, J. (1996), “Mechanical behavior of unsaturated soils: understanding the behavior of unsaturated soils requires reliable conceptual models”, In: Alonso EE and Delage P. (eds), Proceedings of 1st International conference on Unsaturated Soils, Paris, vol. 3, Balkema Presses des Ponts et Chaussees pp. 1223-1256, incorporated herein by reference in its entirety. Conceptual clay structure in FIGS. 3A, 3B and 3C, respectively, represents the fabric on wet and dry side of optimum. However, one of major limitation in Gens and Alonso model is the consideration of the two micro level voids as one. This leads to the presence of two global levels only (intraplatelet spaces and microvoids between platelets) and contradicting the fact that microvoids and intervoid space between platelets may also be present in an unsaturated state. Therefore, their model may only be considered applicable to heavily compacted clays such as ones being used for the nuclear and other types of waste containment. Likos and Lu were the first to consider a more realistic fabric consisting of inter-aggregate, intra-aggregate (or inter-particle) and interlayer space levels. See Likos, W. J. and Lu, N. (2006), “Pore scale analysis of bulk volume change from crystalline swelling in Na+- and Ca2+-smectite”, Clays and Clay Minerals, Vol. 54, No. 4, pp. 516-529, incorporated herein by reference in its entirety. Their model is shown in FIGS. 4A, 4B and 4C.
The clay fabric was also further elaborated by Sharma, defining the micro and macro structure as assemblage of particles with three levels of voids as micro, macro and intra platelet voids. His conceptual model is shown schematically in FIGS. 5A, 5B and 5C. Sanches et al. also adopted the model of Gens and Alonso considering the two general levels of structures (FIG. 6) and the assumption that microstructure is being considered as saturated at all the field conditions. See Sanchez, M., Gens, A Guimaras, L. N. and Olivella, S. (2005), “A double structure generalized plasticity model for expansive materials”, International Journal for Numerical and Analytical Methods in Geomechanics, 2005, 29:751-787, incorporated herein by reference in its entirety. Pinyol et al. studied and modeled the weathering of the soft clayey rocks and also considered a model similar to the Gens and Alonso with the addition of cementation at the platelet contacts. See Pinyol, N., Vaunat, J. and Alonso, E. E. (2007), “A constitutive model for soft clayey rocks that includes weathering effects”, Geotechnique 57, No. 2, 137-151, incorporated herein by reference in its entirety. They also modeled the degradation of the cementation upon cyclic loading and weathering conditions. The conceptual model prepared by them for the cyclic load simulation is shown in FIG. 7. This model could be considered a promising attempt to incorporate natural soil behavior but may be applicable to the homogenous type of clay rocks only and not to the soils consisting of multiple minerals. Moreover, Pinyol et al. did not consider the effects of presence of fissures and cracks present in the natural clay fabric. These fissures and cracks should be considered inherent part of the natural deposits and contribute significantly towards the digenesis and weathering processes.
Fityus and Buzzi discussed and reviewed the effects of the clay microfabric on the volume change of the macrofabric in the existing models. See Fityus, S. and Buzzi, O (2008), “The place of expansive clays in the framework of unsaturated soil mechanics”, Applied Clay Science, Vol. 43, Issue 2, page 150-155, incorporated herein by reference in its entirety. They conceptualized clay structure as a group of aggregates and clusters into single structural element group called peds. A ped is a naturally occurring, structured soil element within a ripened (Pons and Van der Molen) heavy clay soil; that is bounded by discontinuities (typically cracks) that separate it from the adjacent elements of similar form. See Pons, L. J. and Van der Molen, W. H. (1977), “Soil genesis under dewatering regimes during 1000 years of polder development”, Soil Science 116, 228-235, incorporated herein by reference in its entirety. The ped could therefore be considered as basic unit of natural heavy clay soil at the macro scale. Particle size of montmorillonite particle size being in the order of 50 to 1600 nm (Robertson et al.), it becomes difficult to characterize the structure and the pore spaces even using the most advanced and sophisticated Environmental Scanning Electron Microscope (ESEM) and X-ray Computed Tomography (CT) scanning or mercury porosimetry techniques (Fityus and Buzzi). See Robertson, H. E., Weir, A. H. and Woods, R. D. (1968), “Morphology of particles in size fractionated Na montmorillonite”, Clays and Clay Minerals 16, 239-247, incorporated herein by reference in its entirety. Both naturally occurring soils and soils created from the consolidation of slurries have a very small pore size of an order of 3-10 nm and air entry value of 80-100 MPa as reported by Alymore and Quirk, Oades and Waters, Villar, and Meunier. See Alymore, L. A. G. and Quirk, J. P. (1962), “The structural status of clay systems”, In: Swineford, A. (Ed.), Proceedings of the 9th National Conference on Clays and Clay Minerals, Lafayette, Ind., pp. 104-130; Villar, M. V. (2000), “Thermo-hydro-mechanical characterization of a bentonite from Cabode Gata”, PhD Thesis, Universidad Complutense, Madrid, Spain; Meunier, A. (2006) “Why are clays minerals small”, Clay Minerals 41, 551-566, each incorporated herein by reference in their entirety. Based on this fact, saturation of the peds pass through drying and shrinkage cycles without any water loss and complete saturation is ensured at all the field suction values. However, Terzaghi's saturation and effective stress concepts could not be considered applicable to the saturated peds (Lambe and Whitman; Sridharan and Venkatappa; Heuckel. See Lambe, T. W., Whitman, R. V. (1959), “The role of effective stress in the behavior of expansive soils”, First Annual Soil Mechanics Conference, Colorado School of Mines, pp. 33-65; Sridharan, A., Venkatappa R. G. (1973), “Mechanisms controlling volume change of saturated clays and the role of the effective stress concept”, Geotechnique 23, 359-382; Hueckel, T. A. (1992), “Water-mineral interaction in hygromechanics of clays exposed to environmental loads: a mixture-theory approach”, Workshop on Stress Partitioning in Engineered Clay Barriers, May 29-31, 1991, Duke University, Durham, N.C. 1071-1086, each incorporated herein by reference in their entirety. The structure envisioned as saturated soil peds separated by air-filled macroscopic desiccation cracks (FIG. 8) confirms that it cannot be modeled either as continuum or as unsaturated soils due to non-existence of surface films and water bridges.
Likos and Wayllace studied the porosity evolution of free and confined bentonite during the phase of the interlayer hydration. See Likos, W. J., Wayllace, A. (2010), “Porosity Evolution of Free and Confined Bentonites During Interlayer Hydration” Clays and Clay Minerals, Vol. 58 (3), pp. 399-414, incorporated herein by reference in its entirety. They came up with a bimodal porosity model developed for Wyoming bentonite using SEM image of the compacted bentonite. The schematic sketch of the model at several levels is shown in FIGS. 9A, 9B, 9C, 9D, and 9E. Hueckel, while presenting his mixture theory approach for water-mineral interaction in clays under environmental loads provided in a schematic sketch various forms of water in high density clayey soil. His concept of various forms of water and the corresponding pores in a natural soil deposit are shown in FIG. 10.
In addition to the fabric visualization of the expansive clayey soils, another important input required in any molecular level modeling/simulation is the size of the fundamental/smallest clay mineral crystallites. Several researchers have come up with a fundamental size ranging from as small as 100 Å (Longuet-Escard et al.) to much greater than 1000 Å. See Longuet-Escard, J., Mering, J., and Brindley, G. (1960), “Analysis of hk bands of montmorillonite”, C. R. Acad, Sci, Paris 251, 106-108, incorporated herein by reference in its entirety. Most probable reason for such wide range of clay mineral crystallite is the method used for the determination of the size. It has been observed that at most of the times, the imaging or mapping methods involve use of dry specimens. In dry form, the crystallites most probably get fused at the edges and ends and grow into larger crystallites. Moreover, flocculated fabric may also be responsible for such discrepancy. Therefore, the techniques involving the wet specimens such as ESEM and in the dispersed fabric form could provide the real fundamental crystallite size for clay minerals.
The above discussions on the fabric of clays being conceptualized in the existing constitutive models reveal that there are several underlying simplified assumptions that obscure the real behavioral contribution from several levels. This fact is particularly true for the molecular/nano level contributions to macro level behavior. All the existing models ignore the molecular level considerations in their assumed clay fabric and hence its fundamental role in the overall behavior of expansive clays.
Highly charged clay particles/platelets make bonds with water and the dissolved ions to satisfy their charges and consequently an expansion of their structures occurs. These expanded structures have a tendency to collapse i compress shrink upon loss of water. Clay particle-water interaction theories date back to early 20th century when Guoy and Chapman came up with their diffuse double layer (DDL) theory. See Gouy, G. (1910), “Sur la constitution de la charge electrique a la surface d'un electrolyte”, Annales de Physique (Paris), Serie 4, 9, 457-468; Chapman, D. I. (1913), “A contribution to the theory of electrocapillarity”, Philosophical magazine, Vol. 25 (6), 475-481, each incorporated herein by reference in their entirety. This theory was later on further refined by Stern. See Stern S. (1924), “Modification in Diffuse Double Layer Theory”, Z. Elektrochem., Vol. 30, p. 508, incorporated herein by reference in its entirety. In order to satisfy charges, DDL develops for individual clay units and platelets and is schematically shown in FIG. 11. DDL can successively model the effects of cation valence, dielectric constant, electrolyte concentration, and temperature. However, there are certain limitations associated with DDL such as cations are being considered as point charges, DDL may not develop in highly compacted soils and there is less likelihood of presence of parallel clay particles in real clay fabric. Recently, Wayllace developed a general understanding of the structure of the swelling clay minerals, short and long-term water adsorption mechanisms, and influences of particle and pore fabric on swelling behavior using the porosity evolution model developed by Likos and Lu; the model is conceptually shown in FIG. 12. See Wayllace, A. (2008), “Volume change and swelling pressure of expansive clay in the Crystalline swelling regime”, PhD Thesis, University of Missouri, US, incorporated herein by reference in its entirety.
Wayllace divided the water adsorption phenomenon of the clay minerals into three micro-scale mechanisms as hydration, capillarity, and osmosis. Hydration and osmosis play a central role in two main clay swelling processes i.e. crystalline and osmotic swelling (Marshall; Van Olphen; madsen and Muller-Vonmoos). See Marshall, C. E. (1949) “The Colloid Chemistry of the Silicate Minerals”, New York: Academic Press, P. 54; Van Olphen, H. (1977), “An introduction to clay colloid chemistry”, 2nd ed. New York: John Wiley and Sons; Madsen, F. T. and Muller-Vonmoos, M. (1989), “The swelling behavior of clays”, Applied Clay Science 4:143-56, each incorporated herein by reference in their entirety. Capillary mechanism is responsible only for the provision of the water for other major and short-ranged water adsorption mechanisms (Snethen et al.; Miller). See Snethen, D. R., Johnson, L. D. and Patrick, D. M. (1977), “An Investigation of the Natural Microscale Mechanisms That Cause Volume Change in Expansive Clays” Federal Highway Administration Report No. FHWA-RD-77-75; Miller, D. J. (1996) “Osmotic suction as a valid stress state variable in unsaturated soils” Ph.D. dissertation, Colorado State University, Fort Collins, Colo., each incorporated herein by reference in their entirety. Wayllace emphasized the importance of the crystalline or type-I swelling as the key mechanism leading to a better understanding of the swelling behavior. Crystalline swelling is a process whereby expandable 2:1 phyllosilicates sequentially intercalate one, two, three or four discrete layers of H2O molecules between the mineral interlayer (Norrish) shown schematically in FIG. 13. See Norrish, K. (1954), “The Swelling of Montmorillonite”, Transaction Faraday Society 18: pp. 120-134, incorporated herein by reference in its entirety. Type-II swelling mechanism involves the hydration of the cations dissolved in the water layers. For example, Van Olphen calculated that for Ca-montmorillonite, the pressure associated with removing the water from the fourth, third, second, and first hydration states were 20,000 kPa, 125,000 kPa, 250,000 kPa, and 600,000 kPa, respectively. See Van Olphen, H. (1963), “Compaction of Clay Sediments in the Range of Molecular Particle Distances”, Clays and Clay Minerals, Vol. 11, pp. 178-187, incorporated herein by reference in its entirety.
Osmotic theory has also been used to explain the swelling characteristics of the clay particles (Bolt). See Bolt, G. H. (1956), “Physico-chemical Analysis of the Compressibility of Pure Clays”, Geotechnique, Vol. 6, No. 2, pp. 86-93, incorporated herein by reference in its entirety. An equilibrium analysis is carried out between the unit layers, clay platelets, and water by balancing the external and internal forces in order to achieve the maximum number of layers in a platelet. In order to maintain equilibrium, water flows from low concentration (bulk water) to higher concentration of ions (DDL water) and increases the pressure in the DDL. This high pressure in turn causes the tendency to have a reverse flow till a balance is reached.
Few efforts have also been made at nano level to model the swelling mechanism of the swelling clays. The results of these studies are in some cases in contradiction of the general understanding of the swelling clays. This emphasizes the need for nano level modelling and consequently refinement and augmentation of the existing micro and macro scale models.
Swell potential modeling of expansive clays have been carried out by several researchers with an objective of formulating the representative constitutive models. In this regards, efforts have been made at macro, micro, and nano/molecular levels to constitute behavior models for the expansive clays. Most of the constitutive modeling studies have been carried out at macro/micro levels and the simulations have been performed at nano/molecular level.
Constitutive model of expansive clays could be considered as a special case of the general constitutive models for the unsaturated soils. In the realm of unsaturated soils, Matyas and Radhakrishna could be considered as the pioneers to create the concept of state (constitutive) surfaces relating the void ratio and degree of saturation with the state parameters net stress, p and suction, s. See Matyas, E. I. and Radhakrishna, H. S. (1968), “Volume change characteristics of partially saturated soils”, Geotechnique, Vol. 18 (4), 432-448, incorporated herein by reference in its entirety. These surfaces are characterized by one of the very basic observation of the wetting induced swelling at low mean net stress while wetting induced collapse/compression at high mean net stress. The idea of state surfaces was, later on, extended and developed by Fredlund and Morgenstern and Fredlund and was called State Surface Approach (SSA). See Fredlund, D. G. and Morgenstern, N. R. (1977), “Stress state variables for unsaturated soils”, Journal of Geotechnical Engineering Division, ASCE, Vol. 103(GT5), 447-466; Fredlund, D. G. (1979), “Appropriate concepts and technology for unsaturated soils”, Canadian Geotechnical Journal, Vol. 16, 121-139, each incorporated herein by reference in their entirety. The equations suggested by the authors represent the planar surfaces and are limited by the fact that these do not account for the wetting induced collapse and swelling. Moreover these are valid only for monotonic loading and not for wetting and drying cycles. In addition, as stated above, no distinction can be made between elastic and plastic strains as these are only representative of the elastic zones. However, Fredlund (1979) suggested that these relations could be representative of the elasto-plastic strains if constants are functions of stress state. Later on, Lloret and Alonson proposed state surfaces relating void ratio and degree of saturation. See Lloret, A. and Alonso, E. E. (1985), “State surfaces for partially saturated soils”, Proc 11th Conference on Soil Mechanics and Foundation Engineering, Sand Francisco, Vol. 2, 557-562, incorporated herein by reference in its entirety. Although these relations represent surfaces that can simulate the wetting induced compression and swelling behavior but these were again valid only over a limited stress interval.
Alonso et al. (1987) were the first ones to present an integrated volumetric and shear strength elasto-plastic framework of the unsaturated soils. See Alonso, E. E., Gens, A. and Hight, D. W. (1987), “Special Problem Soils, General Report”, Proceedings 9th European Conference on Soil Mechanics, Dublin, Vol. 3, 1087-1146, incorporated herein by reference in its entirety. The qualitative framework was further developed into its mathematical form by Alonso et al. (1990) in their landmark paper and was named Barcelona Basic Model (BBM). See Alonso, E. E., Gens, A. and Josa, A. (1990), “A constitutive model for partially saturated soils”, Geotechnique, Vol. 40(3), 405-430, incorporated herein by reference in its entirety. It would be quite correct to state that all the recent models for unsaturated expansive and non-expansive soils are based on the same core of the BBM. Alonso et al. (1990) provided a complete mathematical formulation of the critical state based model for non-expansive or slightly expansive unsaturated soils. Four state variables i.e., mean net stress, suction, deviator stress, and the specific volume were used to formulate the model. The projection of the yield surface on p-s space (isotropic stress space) is a curved line known as Load-Collapse (LC) curve and shown in FIGS. 14A, 14B, 14C and 15. Plastic compression at high stress level upon wetting is modeled in a similar way as the plastic compression after crossing the yield point and change of specific volume upon plastic yielding. Volumetric decrease as a result of the increase in suction is delimited by the yield surface or limiting line of Suction Increase (SI) shown in FIGS. 15A, 15B, 15C, and 15D. Both SI and LC together mark the area characterized as elastic zone. They used the modified cam clay model as the interface with the saturated counterpart. Therefore, yield surface is an ellipse in anisotropic states in q:p plane at all suctions (FIG. 16). Although non-linearity of the shear strength is well established but for the sake of the simplicity for the initial model, it has been taken as linear. Their proposed shear strength equation collapses to the one proposed by Fredlund (1979) when c′=0. They proposed non-associated flow rule model to match well with the Ko conditions of saturated sand. Ten soil constants are required for the development of the model while current soil state is defined as p, q, s and v or p, q, s and p(0). The model developed by Alonso et al. (1990) is volumetric in nature only and no consideration of mechanical behavior is taken in the model. Simplifying assumptions adopted in the model are the use of straight lines for the e-ln p relationships (implying a continuous increase of the collapse strains upon wetting) and the linear increase of apparent cohesion with suction. Moreover, no hydraulic hysteresis has been incorporated in the model. In spite of its basic nature, BBM was quite able to define several typical behaviors of unsaturated soils such as the variation of wetting-induced swelling or collapse strains depending on the magnitude of applied stresses, the reversal of volumetric strains observed during wetting-induced collapse, the increase of shear strength with the increase in suction, stress path independency associated with wetting paths and the opposite when the stress path involves drying or the apparent increase of pre-consolidation stress with suction. BBM became a basis for its specific and advanced model for the expansive soils, BExM.
As BBM was developed for the non-expansive or slightly expansive soils, Gens and Alonso provided a breakthrough in the provision of a conceptual model encompassing the behavior model for expansive clays. The model was based on the behavior of an extension of the BBM. This model covers the limitation of the BBM to model the large strain behavior of expansive soils and hence introduced a microstructure model to be coupled with the macrostructure model of Alonso et al. (1990). In their coupled models, soil structure has been divided into two distinct levels i.e. micro and macro. Microstructure consists of quasi-crystals, particles assemblages, and pore spaces, while assemblages together formulate matrix in which large sized sand and silt particles are embedded. The extended model incorporates a microfabric of clay particles and aggregations embedded into an overall macrofabric of silt and sand size particles. The elementary particles group together to form aggregations and resulting in granular type of structure. The pores sizes in the formulated structure are present both as intra and inter aggregations. They considered microfabric to be only affected by the local stresses and hence effective stress principles may be applicable and volume change in microfabric to be reversible and unaffected by strain in the macrofabric. This assumption leads to the fact that if sum of net stress and suction (p+s) remains constant, then no change in overall volume would occur and the stress state moves on a line known as neutral line (FIG. 17). The microfabric in their model is essentially considered saturated even if the overall saturation of the soil fabric is not achieved. Although, micro structural level behavior remains generally independent of the macrostructure behavior and is basically controlled by the physicochemical processes causing volume variations, there is an obvious interaction and this has been covered in the extended model by coupling of the micro and macro structure (FIGS. 18A and 18B). Therefore, the extended model for expansive clays should consist of three elements as soil behavior at macroscale, behavior at microscale, and the coupling between the two levels. One of the major limitations of the Gens and Alonso model was the assumption regarding the permanent saturation of the microfabric as that does not seem to be realistic as microvoids/inter platelet voids may remain unsaturated as well. The permanent saturation of the microfabric may be considered valid only for the intraplatelet fabric only. Moreover, this model was mainly conceptual in nature and no detailed mathematical formulation was provided till a complete mathematical model by Alonso et al. (1999), named as Barcelona Expansive Model (BExM). See Alonso, E. E Vaunat, J., Gens, A. (1999), “Modeling the mechanical behavior of expansive Clays”, Eng Geol 1999; 54:173-83, incorporated herein by reference in its entirety.
Up to this stage, it is clear that modeling of expansive clays require consideration of three basic elements: microstructure model, macrostructure model and the interaction in the form of coupling functions. From Gens and Alonso model onwards, both the micro and macro level models of the unsaturated soils were mostly handled independently by several researchers. However, most of the researchers worked towards the development and improvement of the unsaturated soil model for non-expansive soils, while only few accomplished some improvements and variations in the expansive clays model.
Alonso et al. (1999) had a landmark contribution in expansive clays model by developing a mathematical model for expansive clays based on the concepts developed in the models of Alonso et al. (1990) and Gens and Alonso. Two additional yield surfaces, one for plastic yielding caused by suction increase (SI) and the other by suction decease (SD), were introduced (FIGS. 19 and 20). These surfaces are parallel to the neutral loading line in the space of net mean stress versus suction, and are coupled to the LC surface through two experimentally determined functions. The model by Alonso et al. (1999) is able to predict the irreversible expansion caused by wetting at low stresses and shrinkage at high stresses. In this model, macro-structural plastic volumetric change causes a corresponding change in the location of the LC. When the macro-structure becomes looser, the macro-structural yield surface shrinks. When the structure becomes denser, the elastic domain increases and LC expands. A coupling therefore exists between yield surfaces LC, SI and SD (FIGS. 19 and 20). However, irreversible change of degree of saturation during cyclic wetting and drying was not considered in the model of Gens and Alonso or of Alonso et al. (1999) and this has remained one the major limitation of these models even to the present.
After Alonso et al. (1999) BExM, major contribution towards the development of the expansive clay models was done by Sanchez et al. who formulated an expansive clays model considering concepts of classical and generalized plasticity theories and is shown in FIGS. 21A and 21B. They developed generalized stress-strain rate equations from the concept of a framework of multi-dissipative materials. This framework provides a consistent and formal approach when there are several sources of energy dissipation and is well suited for the modeling of generalized stress reversals. They used a generalized plasticity model for the materials that show irrecoverable deformations upon reloading and also to include the behavior of soils under cyclic loading when they exhibit irreversible deformation in loading, unloading, and reloading. They were successful in modeling the typical aspects of the behavior observed in expansive soils under generalized stress paths including suction and stress changes. The authors attributed significant advantages in using generalized plasticity theory to model the plastic mechanism related to the interaction between two levels of pores structures.
Sanchez et al. formulated the model in the space of stresses, suction and temperature; and implemented the double structure approach in a finite element program CODE BRIGHT. The mechanical law of this model is able to model the macropore invasion induced by microstructure expansion, when conditions of high confinement prevail considering negative values of the function fs for high values of p=po (FIGS. 21 and 22). In FIG. 22, point at which both interaction curves meet, indicated as E; is the equilibrium point. This point represents the state of the material for which no cumulative deformations are observed after cycles of suction changes.
Next major contribution in the modeling of expansive clays could be considered by Pinyol et al. who investigated the dual nature of Claystone by developing independent constitutive models for their rock-like and clay-like behavior. Claystone acts like a Rock when present in the unweathered state while it behaves as Soil in its weathered state. The authors attributed this dual behavior to the presence of basic clay matrix and the quasi-brittle cementation at the microstructure level. They considered the matrix behavior by the elasto-plastic double structure model proposed by Gens and Alonso and Alonso et al. (1999), while cementation/bonding was modeled using the damage mechanics based model. They demonstrated the effectiveness of the developed integrated models through the data generated through experimentation. Models by Pinyol et al. could also be considered a substantial contribution in modeling the natural homogenous types of clay.
The challenge with BExM is that the micro parameters and the function coupling the micro and macro structural strains are difficult to determine experimentally. Moreover, BExM is mainly concentrating on the stress-strain and strength behavior without considering the water retention behavior of the expansive soils. In this respect, several models have been developed for unsaturated soils but no such effort has been made for expansive soils. Sun and Sun developed an elasto-plastic constitutive model for predicting the hydraulic and mechanical behavior of unsaturated expansive soils based on an existing hydro-mechanical model for unsaturated non-expansive soils. See Sun, W. and Sun, D. (2011), “Coupled modeling of hydro-mechanical behavior of unsaturated compacted expansive soils”, International Journal of Numerical and Analytical Methods in Geomechanics, Vol. 36, Issue 8, page 1002-1022, incorporated herein by reference in its entirety. They basically developed the first macroscopic elastoplastic model for unsaturated expansive soils and also introduced the concept of equivalent void ratio curve to distinguish between the yield curve and plastic potential curve. Basis is the experimental data and the model developed for unsaturated non-expansive soils. This model incorporates the coupled hydro-mechanical effect of degree of saturation on the mechanical behavior and void ratio on the water-retention behavior. Sun and Sun argued that compression index of swelling clays have been found to be increasing with increase in suction while it decreases with increase in suction for unsaturated non-expansive soils. This is a fundamental difference among the compressibility behavior of the unsaturated non-expansive and expansive soils. Their hydro-mechanically coupled elastoplastic model can predict the hydraulic and mechanical behavior of unsaturated expansive soils. While developing that model, they assumed that pore air and pore-water are continuous throughout the soil voids which are basically true for some regime of water content (degree of saturation) only. Besides being a macroscopic model, this is in fact a major limitation of the model.
Guimaraes et al. may be considered as the pioneer in the formulation of a chemo-mechanical model for the expansive clays with due consideration of the contribution from cation content, osmotic suction, and the cation exchange. See Guimares, L. D., Gens, A., Sanchez, M., and Olivella, S. (2013), “A chemo-mechanical constitutive model accounting for cation exchange in expansive clays”, Geotechnique 63, No. 3, 221-234, incorporated herein by reference in its entirety. Their model is a contribution to the microstructure model in the double-structure approach used by Sanchez et al. Their main assumption regarding the elastic or reversibility of the microstructure behavior remains the same. They introduced additional parameters for the microstructure to be incorporated into the constitutive model. Although, the model is quite capable of predicting the behavior of saturated and unsaturated behavior, but most of the basis is through the indirect inferences from macro level studies and no input from molecular level has been incorporated.
Expansive clay minerals are nano-materials and nano-mechanics concepts can be used to improve fundamental understanding of the behavior and predict the volumetric changes under the desired boundary and stress conditions. By obtaining molecular-scale material properties, the macro-scale material behavior can be obtained, with limited input parameters and with great accuracy and details.
For the purpose of molecular/nano level simulations, most commonly adopted technique is the Molecular Dynamics (MD). MD is a computational method which calculates the time dependent behavior of a molecular system. MD is based on Newton's second law of motion and provides a trajectory which specifies the variation of position and velocity of individual atoms in a molecular system with time. In this technique, Individual atoms are characterized by balls with bonds represented as springs. A variety of springs are introduced that capture stretching, angular rotation, and torsion non-bonded interactions are modeled as van der Waals and electrostatic. In MD, individual atoms would be represented by the balls and the connecting major bonds as springs, while non-bonded interactions among the molecules would be represented by the van der Waal's and electrostatics. The potential energy of the system is then calculated using a force-field and is used to calculate the trajectory of the atoms in a molecular system. Force-field (Brooks et al.) is generally given by (see Brooks, B. R., Bruccoleri, R. E., Olafson, B. D., States, D. J., Swaminathan, S., Karplus, M. (1983), “CHARMM: A program for macromolecular energy, minmimization, and dynamics calculations”, J. Comp. Chem. 4, 187-217, incorporated herein by reference in its entirety):
                                          E            Total                    =                                    E              coul                        +                          E              VDW                        +                          E                              Bond                ⁢                                                                  ⁢                Stretch                                      +                          E                              Angle                ⁢                                                                  ⁢                Bend                                      +                          E              Torsion                                      ⁢                                  ⁢        where        ⁢                                  ⁢                              E            coul                    =                                                    e                2                                            4                ⁢                                  πɛ                  o                                                      ⁢                                          ∑                                  i                  ≠                  j                                            ⁢                                                                    q                    i                                    ⁢                                      q                    j                                                                    r                  ij                                                                    ⁢                                  ⁢                              E            VDW                    =                                    ∑                              i                ≠                j                                      ⁢                                          D                o                            ⁡                              [                                                                            [                                              Ro                                                  r                          ij                                                                    ]                                        12                                    -                                                            2                      ⁡                                              [                                                                              R                            o                                                                                r                            ij                                                                          ]                                                              5                                                  ]                                                    ⁢                                  ⁢                              E                          Bond              ⁢                                                          ⁢              Stretch                                =                                                    k                1                            ⁡                              (                                  r                  -                                      r                    o                                                  )                                      2                                              2        ⁢                  -                ⁢        1            
Skipper et al. performed the swelling simulation of various clay minerals using Monte Carlo (MC) simulation technique. See Skipper, N. T., Sposito, G., and Chang, F. R. (1995a), “Monte Carlo simulations of interlayer molecular structure in swelling clay minerals 1. Methodology”, Clays and Clay Minerals, Vol. 43, No. 3, pp. 285-293, Skipper, N. T., Sposito, G., and Chang, F. R. (1995b), “Monte Carlo simulations of interlayer molecular structure in swelling clay minerals 1. Monolayer Hydrates”, Clays and Clay Minerals, Vol. 43, No. 3, pp. 294-303, each incorporated herein by reference in their entirety. They used MONTE, (Skipper) software for the purpose. See Skipper, N. T. (1992), “MONTE User's Manual”, Technical Report, Department of Chemistry, University of Cambridge, UK, incorporated herein by reference in its entirety. They explained the methodology and the simulation details in two of their consecutive papers (Skipper et al.), respectively. They defined the atomic positions and the corresponding effective charges of the clay minerals for the simulation purpose. The outcome of the study showed that Monte Carlo simulations of the Wyoming-type montmorillonite and vermiculite have resulted in layer spacings, average potential energies, and molecular structure that are consistent with the experimental findings.
Karaborni et al. was one of the early researchers who adopted MD for the nano level simulations. See Karaborni, S., Smit, B., Heidug, W. Urai, E. and van Oort, (1996), “The Swelling of Clays: Molecular Simulations of the Hydration of Montmorillonite”, Science, Vol. 271, 23 Feb. 1996, 1102-1104, incorporated herein by reference in its entirety. They performed molecular dynamics and Monte Carlo simulations to study the lattice expansion mechanism of the Na-montmorillonite (FIG. 23). The simulation results revealed and confirmed the generally accepted theory of four stable states at lattice basal spacings of 9.7, 12.0, 15.5 and 18.3 Å respectively. They also proved that swelling percentages and the swelling sites in the stable form of the Na-montmorillonite are generally in good quantitative agreement with the previous studies. The swelling process resulted in the development of one, three, and then five water layers. This anomalous behavior has been found to be contradicting the general concept of formation of hydrated cations layers of one, two, three, four etc. in Na-montmorillonite. They also theorized that relative amount of water adsorbed by Na-montmorillonite is a result of the balance between the hydrogen bonding between water and the tetrahedral sheets of the clay and water adsorption in the clay hexagonal cavities. Based on this theory, they defined the stable states to be those where one of the interaction becomes dominant, while an unstable state to be one where a ‘frustration effect’ is created due to the predominance of both the phenomena simultaneously. They attributed higher swelling potential of Na-montmorillonite to this phenomenon. Therefore, a transition would be required from one orientation of water molecules to a second in order to cause expansion to the clay structure. Clearly, this transition requires lesser free volume of water and is quite easy to take place than the one with the addition of the simultaneous complete layers of water molecules.
Katti et al. (2005) conducted Molecular Dynamics (MD) study of the interlayer response of pyrophyllite under the influence of water and cations in the interlayer. See Katti, D. R., Schmidt, S., Ghosh, P., and Katti, K. S., (2005), “Modeling Response of Pyrophyllite Clay Interlayer to Applied Stress Using Steered Molecular Dynamics”, Clays and Clay Minerals, Vol. 52, n2, 171-178, incorporated herein by reference in its entirety. They used NAMD (Phillips et al.) and VMD software to perform interactive simulations and these were simulated on the North Dakota State University 32 processor parallel computer system. See Phillips, J. C Braun, R., Wang, W., Gumbart, J., Tajkhorshid, E., Villa, E., Chipot, C., Skeel, R. D., Kale, L., and Schulten, K (2005), “Scalable molecular dynamics with NAMD”, Journal of Computational Chemistry, 26(16), 1781-1802, incorporated herein by reference in its entirety. One of the major parts of the study was to transform the Consistent Force Field (CFF) parameters earlier developed by Teppen et al. to CHARMm force field parameters. See Teppen, B. J., Rasmussen, K., Bertsch, P. M., Miler, D. M., and Schafer, L. (1997), “Molecular dynamics modeling of clay minerals. 1. Gibbsite, kaolinite, pyrophyllite, and beidellite”, Journal of Physical Chemistry B, 101, 1579-1587, incorporated herein by reference in its entirety. These were later on used with the NAMD software. Basic pyrophyllite model and the force applied model developed by the authors are respectively shown in FIGS. 24 and 25. In this study, forces were applied on the clay surfaces ranging from 0 pN to 160 pN simulating an equivalent stresses of 0 to 1.65 GPa. The authors concluded that deformation of the clay layers observed in this stress range is only ˜1.6% compared to ˜12.9% for the interlayer. The modulus of the interlayer and the two-clay-layer unit were found to be 13.18 GPa and 54.56 GPa, respectively.
Wang et al. (2007) studied the elastic properties of several minerals including quartz, albite, calcite, montmorillonite, kaolinite and palygorskite through MD technique. See Wang, J., Sharma A. and Gutierrez, S. M. (2007), “Nanoscale Simulations of Rock and Clay Minerals”, ASCE Geotechnical Special Publication 173: Advances in Measurement and Modeling of Soil Behavior Geo-Denver 2007: New Peaks in Geotechnics, incorporated herein by reference in its entirety. They modeled these minerals using both bonded and non-bonded interatomic contributions. The interatomic bonding energies, used in the molecular simulation, are expressed in the following Newtonian form as below:
                                          m            i                    ⁢                                                    d                2                            ⁢                              r                i                                                    dt              2                                      =                  F          i                                    2        ⁢                  -                ⁢        2            
The force F1 acting on a particle i is calculated from the interatomic potential function U (r, r1, r2, rN . . . )
                                          F            i                    =                                                                      ∂                                      U                    ⁡                                          (                                                                        r                          i                                                ,                                                  r                          2                                                ,                                                  …                          ⁢                                                                                                          ⁢                                                      r                            N                                                                                              )                                                                                        ∂                                      r                    i                                                              .                                                          ⁢              i                        =            1                          ,        2        ,                  …          ⁢                                          ⁢          N                                    2        ⁢                  -                ⁢        3            
Dynamics of the system is dominated only by the interatomic potential function U that is representative of the atomic interaction owing to the complex quantum effects occurring at the subatomic level. They utilized the most commonly adopted pair-wise potentials inclusive of Lennard-Jones (LJ) and Morse potentials, as in the following equations:
                                          U            ⁡                          (                                                r                  i                                ,                                  r                  j                                            )                                =                                    U              ⁡                              (                r                )                                      =                          4              ⁢                              ɛ                ⁡                                  [                                                                                    (                                                  σ                          r                                                )                                            12                                        -                                                                  (                                                  σ                          r                                                )                                            6                                                        ]                                                                    ,                                  ⁢                  r          =                                                                  r                ij                                                    =                                                                                                r                    i                                    -                                      r                    j                                                                              .                                                          ⁢                              (                                  LJ                  ⁢                                                                          ⁢                  potential                                )                                                                        2        ⁢                  -                ⁢        4                                          U          ⁡                      (            r            )                          =                  ɛ          ⁢                                    ⌊                                                e                                      2                    ⁢                                          β                      ⁡                                              (                                                  ρ                          -                          r                                                )                                                                                            -                                  2                  ⁢                                      e                                          β                      ⁡                                              (                                                  ρ                          -                          r                                                )                                                                                                        ⌋                        .                                                  ⁢                          (                              Morse                ⁢                                                                  ⁢                potential                            )                                                          2        ⁢                  -                ⁢        5            
The potential function used by Sato et al. and Ichikawa et al. for the simulation of several clay minerals was used to simulate the specific minerals. See Sato, H., Yamagishi, A. and Kawamura, K. (2001), “Molecular simulation for flexibility of a single clay layer”, Journal of Physics Chemistry, vol. B 105, 7990-7997; Ichikawa, Y Kawamura, K., Fuji, N. and Nattavut, T. (2002), “Molecular dynamics and multiscale homogenization analysis of seepage-diffusion problem in bentonite clay”, International Journal of Numerical Methods in Engineering 2002; 54:1717-1749, each incorporated herein by reference in their entirety. The function is composed of several potentials such as Coulomb (attractive or repulsive), Born-Mayer-Higgins short range repulsion, van der Waals, and Morse terms. They used TINKER software Ponder for carrying out MD simulations. See Ponder, J. W. (2011), http://dasher.wustl.edu/, Washington University, US, incorporated herein by reference in its entirety. Data input included the initial configuration of the atomic structures and the interatomic potentials assigned to the specific mineral. An NPT (constant number of particles N, pressure P, and temperature T) ensemble was used to acquire the stress-strain behavior of the simulated minerals. The results of the simulations as shown in FIG. 26 reveal a general agreement between the measured and known values of modulli for the minerals except Kaolinite. The authors have attributed the anomalously higher modulus value of Kaolinite to the molecular arrangement at the crystal lattice level.
Wang and Gutierrez (2007) conducted a molecular simulation study of dehydrated 2:1 clay minerals by changing the MD cell size and shape under the general applied stress conditions. See Wang, J., Sharma A. and Gutierrez, S. M. (2007), “Nanoscale Simulations of Rock and Clay Minerals”, ASCE Geotechnical Special Publication 173: Advances in Measurement and Modeling of Soil Behavior Geo-Denver 2007: New Peaks in Geotechnics, incorporated herein by reference in its entirety. The molecular simulation method adopted by the authors considered the basic relationship between the atomic level stress tensors, including internal, external, and the simulation stress tensor. They thoroughly investigated the relaxation behavior of the dehydrated mica sheets by the incorporation of varying boundary conditions on the simulation cell. It was concluded that the degree of freedom of the simulation cell is directly related to the formation of the final crystal structure. One of the important conclusions was the shear deformation of the crystal structure in the absence of any boundary constraint. They also showed that the interlayer spacing could either be reduced or completely removed by application of the high normal pressures.
Katti et al. (2009) studied the effect of swelling and swelling pressure of the montmorillonite clay using the experimental set up and further validated the results using numerical techniques. See Katti, D. R., Matar, M. I., Katti, K. S. and Amarasinghe, P. M. (2009), “Multiscale Modeling of Swelling Clays: A Computational and Experimental Approach”, KSCE Journal of Civil Engineering (2009) 13(4): 243-255, incorporated herein by reference in its entirety. They used a specially designed swelling device to control the swelling and swelling pressure of the sample and studied the clay fabric created at each specified level. They concluded that there is breakdown of the clay particles/assemblages as the swelling of the clay particles increases as a result of intake of water. They used Fourier Transform Infrared Spectroscopy (FTIR) and X-ray diffraction (XRD) techniques to study the microstructure of the swollen clays. They also used Discrete Element Method (DEM) and Steered MID based numerical techniques to model the swelling behavior of clay soils. Basic model of Na-montmorillonite with 3 water layers is shown in FIG. 27, while the plots of stress vs. interlayer strain with the variation of water content is shown in FIG. 29. Based on the experimental and numerical simulation, main conclusion of their study was there is increase in d-spacing of the clay particles as a result of the swelling and beyond certain d-spacing, particle assemblage breakdown takes place and more and more particles are exposed to swelling.
Tao et al. performed molecular dynamics simulations to investigate the role of the cations K, Na, and Ca on the stability and swelling of montmorillonite. See Tao, L Xiao-Feng, T., Yu, Z. and Tao, G. (2010), “Swelling of K+, Na+ and Ca2+-montmorillonites and hydration of interlayer cations: a molecular dynamics simulation”, Chin. Phys. B Vol. 19, No. 10 (2010), incorporated herein by reference in its entirety. They used CLAYFF force field (Cygan et al.) to predict the basal spacing as a function of the water content in the interlayer. See Cygan, R. T., Liang, J. J. and Kalinichev, A. G. (2004), “Molecular Models of Hydroxide, Oxyhydroxide, and Clay Phases and the Development of a General Force Field”, J. Phys. Chem. B 108 1255, incorporated herein by reference in its entirety. All MD simulations were carried out using the LAMMPS software package (Plimpton). See Plimpton, S. J. (1995), “Fast Parallel Algorithms for Short-Range Molecular Dynamics”, J Comp Phys., 117, 1-19, incorporated herein by reference in its entirety. The results of the simulations showed that the swelling pattern of these simulated Montmorillonite is different than that by the corresponding K+, Na+, and Ca2+ montmorillonite (FIG. 29). The authors discovered that Ca-montmorillonite exhibits less swelling than Na- and K-montmorillonite for a given water content. The results of this study also showed that the higher the hydration energy of the interlayer cation, the greater is this difference. In particular, these results indicated that the valence of the cations has the larger impact on the behaviour of clay-water systems.
Katti et al. (2011) presented the results of modeling of molecular interactions between swelling clay and fluids and their effects on the mechanical and flow characteristics. In this study, MD simulations were conducted to study the possible interactions among clay, water, and cations present in the interlayer using MD based software NAMD (Phillips et al.) and the visualization software VMD (Humphrey et al.). See Humphrey, W., Dalke, A., and Schulten, K (1996), “VMD: Visual molecular dynamics”, Journal of Molecular Graphics, 14(1), 33-35, incorporated herein by reference in its entirety. The results of the study showed an increased breakdown of the aggregated particles and their corresponding contributions towards the enhanced swelling and the swelling pressure. Generally, their results showed an agreement with the well-established and determined concepts related to the swelling mechanism of the clay minerals. They discovered that the fact that the forces among clay sheets and Na+ cations are attractive in nature in the dry state. As per their results, these attractive forces/interactions among Na+ and clay surfaces are quite pronounced even up to the presence of 8 water layers in the interlayer and water is still contributing to the attractive forces among hydrated Na+ cations and the water bounded to the clay surfaces even more than 8 water layers (FIG. 30).
Based on the deliberations above, it could be inferred that nano or molecular level processes play a central role in the understanding of the volume change behavior of the expansive clays. Although some studies have been conducted to simulate the swelling and/or water absorption behavior of the single or isolated expansive clay minerals, modeling of the real/natural expansive soil fabric and its interaction with pore fluids at molecular level is still lacking. Moreover, no efforts have been directed to couple the macro and micro scale material behavior to the findings of these molecular simulations. Based on all the above deliberations on the modeling of expansive clay soils, the following may be said:                More effort and emphasis has been directed to the development and enhancement of the unsaturated non-expansive soils models, while much lesser effort has been made towards expansive clay modeling.        Almost all the researchers involved in the unsaturated soils research have considered expansive soils as an extreme case of the unsaturated soils; rather it should be considered both as a special case of saturated soils and unsaturated soils under the complete moisture regime.        Micro and nano level fabrics, believed to have a central role in the overall behavior of expansive clays, are only partially considered in the modeling concepts. Even the partial consideration of the micro and nano level fabric is for the clays compacted/constructed under highly controlled conditions; natural clay fabrics with multiple clay minerals, silt and sand inclusions, micro fissures, cementation, over-consolidation, induration, and other such features have never been considered.        The boundary between expansive and non-expansive soils is not well defined, and more consideration is needed to better understand the behavior of slightly expansive clay soils such as broadly graded soils with small Smectite (and Illite) contents and soil dominated by non-swelling clays such as Kaolinite.        Molecular level research, at present, has just concentrated mostly on one mineral only; interaction with other minerals and macro particles is lacking.        It has been observed that the macroscopic behavior of clay mass may differ considerably from their nano-scale response, which is the major motivation for characterizing and modeling these materials using multiscale simulations.        